UNIPLANAR MOTION OF A RIGID BODY 221
where 7 is the common angular acceleration. Therefore the angular kinetic reaction varies directly as the product of the moment of inertia by the angular acceleration, angular kinetic reaction = kly,
where k is the constant of proportionality. When all the magnitudes involved in the last equation are measured in the same system of units k becomes unity.
Introducing this simplification in the last equation and putting it into vector notation we have
angular kinetic reaction = — JY. (IV)
The negative sign indicates the fact that the direction oi the angular kinetic reaction is opposed to that of the angular acceleration.
184. Torque Equation. — Combining equations (I) and (IV) and denoting the resultant torque by G we obtain
(V)
= ico.
The last equation, which will be called the torque equation, states that the resultant torque about any axis equals the product of the moment of inertia by the angular acceleration and has the same direction as the angular acceleration. 185. The Two Definitions of Moment of Inertia* — In order to show that the constant, I, of equation (II) and the moment of inertia defined by equation (II) of page 152 are the same magnitude, consider the motion of the rigid body A, Fig. Ill, about a fixed axis through the point 0, perpendicular to the plane of the paper. Let dF be the resultant force acting upon an element of mass dm, that is, the vector sum of the forces due to external fields of force and the forces due to the connection of dm with the rest of the body. Then
dF^dm^ at
is the force equation for the element of mass.